General Relativity for the Gifted Amateur

Stephen J. Blundell
University of Oxford

Great Clarendon Street, Oxford, OX2 6DP, United Kingdom
Oxford University Press is a department of the University of Oxford. It furthers the University's objective of excellence in research, scholarship, and education by publishing worldwide. Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries
© Tom Lancaster and Stephen Blundell, 2025
The moral rights of the authors have been asserted
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, transmitted, used for text and data mining, or used for training artificial intelligence, in any form or transmitted, used for text and data mining, or used for training artificial intelligence, in any form or
by any means, without the prior permission in writing of Oxford University Press, or as expressly by any means, without the prior permission in writing of Oxford University Press, or as expressl
permitted by law, by licence or under terms agreed with the appropriate reprographics rights
permitted by law, by licence or under terms agreed with the appropriate reprographics rights
organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above

You must not circulate this work in any other form
and you must impose this same condition on any acquirer
Published in the United States of America by Oxford University Press 198 Madison Avenue, New York, NY 10016, United States of America

British Library Cataloguing in Publication Data
Data available
Library of Congress Control Number: 2024946925
ISBN 9780192867407
ISBN 9780192867414 (pbk.)
DOI: 10.1093/oso/9780192867407.001.0001
Printed and bound by CPI Group (UK) Ltd, Croydon, CR0 4 YY
Cover image: The authors
Links to third party websites are provided by Oxford in good faith and for information only. Oxford disclaims any responsibility for the materials contained in any third party website referenced in this work.

I saw Eternity the other night
Like a great Ring of pure and endless light, All calm as it was bright,
And round beneath it, Time in hours, days, years Driv'n by the spheres
Like a vast shadow mov'd. In which the world
And all her train were hurl'd.
Henry Vaughan (1621-1695) The World
You sometimes speak of gravity as essential and inherent to matter. Pray do not ascribe that notion to me, for the cause of gravity is what I do not pretend to know and therefore would take more time to consider of it.
Sir Isaac Newton (1642-1726) Letter to Richard Bentley
Albert Einstein's crowning theoretical achievement was his formulation of his general theory of relativity, a theory of gravity that superseded Isaac Newton's approach and transformed our view of the Universe. It took a century for one of its key predictions, the existence of gravitational waves, to be verified, but it is a measure of how persuasive its underlying principles have been that no-one seriously doubted that gravitational waves would eventually be detected. General relativity engages profoundly with the nature of space and time (even more so than Henry Vaughan did in his magnificent poem quoted above) and provides the key ideas missing from Newton's theory (the deficiency of which Newton was keenly aware, as indicated in the second quotation). Our point of view in writing this book is that everyone should have the opportunity to engage with this beautiful theory which, conceptually, is based on simple ideas from the physics of fields. The mastery of the machinery of general relativity does however require some facility with mathematics that is likely to be unfamiliar to many students of physics, but the payoff is so great and the material so stimulating that we hope the reader will join us in exploring one of the greatest achievements in physics.
The text follows the same approach as our earlier Quantum Field Theory for the Gifted Amateur (QFTGA) (OUP, 2014) and this is perhaps a good moment to restate what we mean by the slightly tongue-incheek term 'gifted amateur'. We are not writing for the mathematically uninitiated and do assume that our reader has a background in physics. However, we are not writing for experts either and aim to provide an entry point to a profound topic that we hope readers will find both entertaining and useful. The use of the term 'gifted amateur' encourages
the potential reader to have a go for themselves, conveying the feeling that a difficult subject is open to those who considered themselves nonexperts. We adopt the same approach used in writing QFTGA, dividing the material into short and easily digestible chapters, spelling out mathematical steps in worked examples and illustrating the arguments with hand-drawn figures. In response to reader requests from QFTGA, we also include a large number of problems with worked solutions. We have both had a long-standing fascination with the subject and a conviction that it belongs more centrally in the physics curriculum. However, we have not lost the memory of finding some of this material difficult and so hope that our book will give a curious reader a more patient and illuminating guide to this subject that they would find in some of the more weighty and established tomes.
The first two parts of the book introduce the main concepts that lead to the formulation of the Einstein field equations, and this material concludes with an outline description of the most important implications of the theory. These implications are worked out in much more detail in the middle section of the book, the third part covering cosmology and the fourth part detailing the consequences for orbits and black holes. General relativity is a theory about the geometry of spacetime and a more mathematical treatment of geometry is given in the fifth part of the book for those with an appetite to explore these aspects in more detail. The final part of the book returns to field theory, framing general relativity as a classical field theory and looking forward to how it might be formulated as a quantum field theory in a future theory of quantum gravity.
In writing this volume we are particularly grateful to the following individuals who have helped us: Rodrigo Alonso, Nathan Bentley, Katherine Blundell, Theo Breeze, Harvey Brown, Andrei Constantin, Felix Flicker, Martin Galpin, Matjaž Gomilšek, Thomas Hicken, Ben Huddart, Ifan Hughes, Baojiu Li, Guillaume Mahler, and Trevor Wishart. These individuals have been generous with their time and have helped improve the book, but any errors that are found post-publication will be posted on the book's website:
http://tomlancaster.webspace.durham.ac.uk/grgabook
We are also grateful to our copy editor Aravind Kannankara. Finally, we thank Cally, Eden, and Katherine for their patience, love, and support.

TL & SJB

0 Overture1
0.1 What is relativity?1
0.2 What is general relativity?3
0.3 What is a metric? ..... 5
0.4 What are we building on? ..... 6
0.5 Who is this book for?8
0.6 Units in this book
10
Exercises
11
I Geometry and mechanics in flat spacetime
1 Special relativity ..... 12
1.1 A common sense start ..... 12
1.2 The speed of light ..... 13
1.3 Light cones and the Lorentz transformation ..... 15
1.4 Paths through spacetime ..... 18
1.5 Experiments ..... 19
Exercises ..... 21
2 Vectors in flat spacetime ..... 22
2.1 Vectors ..... 23
2.2 Coordinate transformations ..... 24
2.3 Examples of vectors ..... 27
2.4 Principle of least action
34
Exercises ..... 34
3 Coordinates ..... 36
3.1 Coordinates in Euclidean space ..... 36
3.2 Farewell to the position vector ..... 39
3.3 Non-Euclidean space ..... 40
Exercises ..... 41
4 Linear slot machines ..... 43
4.1 Dot products and down vectors ..... 44
4.2 Vectors and 1-forms ..... 46
4.3 Transformations ..... 49
4.4 Tensors ..... 50
4.5 Energy-momentum tensor ..... 52
Exercises ..... 55
5 The metric ..... 56
5.1 Metrics in general ..... 56
5.2 Meet some metrics ..... 58
5.3 Light and light cones ..... 60
5.4 Lengths, areas, volumes ..... 62
Exercises ..... 65
II Curvature and general relativity ..... 67
6 Finding a theory of gravitation ..... 68
6.1 Free fall and the equivalence principle ..... 68
6.2 Why general relativity? ..... 73
6.3 A differential equation to describe gravity ..... 75
6.4 Local flatness ..... 76
6.5 Time dilation in a gravitational field ..... 77
Exercises ..... 79
7 Parallel lines and the covariant derivative ..... 81
7.1 Parallelism ..... 82
7.2 Derivatives and connections ..... 83
7.3 The covariant derivative ..... 85
7.4 Parametrized paths ..... 86
7.5 Enter the metric ..... 88
Exercises ..... 89
8 Free fall and geodesics ..... 91
8.1 Extremal intervals ..... 91
8.2 A geodesic equation ..... 95
8.3 Inertial forces ..... 96
8.4 Geodesics for photons ..... 98
Exercises ..... 99
9 Geodesic equations and connection coefficients ..... 101
9.1 Finding connection coefficients ..... 101
9.2 The geodesic equation from the action ..... 104
Exercises ..... 105
10 Making measurements in relativity ..... 108
10.1 Observers and their observations ..... 108
10.2 Coordinate and non-coordinate bases ..... 110
10.3 The orthonormal frame ..... 114
10.4 Freely falling frames ..... 116
Exercises ..... 118
11 Riemann curvature and the Ricci tensor ..... 120
11.1 What is curvature? ..... 120
11.2 Tidal forces ..... 121
11.3 Riemann curvature ..... 124
11.4 Symmetries of the Riemann tensor ..... 126
11.5 The Ricci tensor and Ricci scalar ..... 127
11.6 Example computations ..... 128
11.7 Geodesic deviation revisited ..... 129
Exercises ..... 130
12 The energy-momentum tensor ..... 131
12.1 Another look at the energy-momentum tensor ..... 131
12.2 Example energy-momentum tensors ..... 132
12.3 Classical particles ..... 134
12.4 Conservation laws ..... 136
Exercises ..... 140
13 The gravitational field equations ..... 141
13.1 Geometry: a recap of the key ingredients ..... 141
13.2 Physics: the key ingredients ..... 142
13.3 An incorrect guess ..... 145
13.4 Einstein's field equation ..... 147
Exercises ..... 150
14 The triumphs of general relativity ..... 151
14.1 Weak fields and the Newtonian limit ..... 151
14.2 Gravitational waves ..... 153
14.3 Stars, trajectories, and orbits ..... 155
14.4 Cosmology ..... 156
III Cosmology ..... 157
15 An introduction to cosmology ..... 158
15.1 The cosmological principle ..... 159
15.2 The Hubble flow ..... 160
15.3 Cosmic time ..... 162
15.4 Universe 0 : an empty universe ..... 163
15.5 Universe 1: flat and expanding ..... 164
Exercises ..... 167
16 Robertson-Walker spaces ..... 169
16.1 Spaces with constant curvature ..... 169
16.2 Three Robertson-Walker spaces ..... 173
16.3 Redshift and cosmic expansion ..... 176
16.4 The initial singularity ..... 178
Exercises ..... 179
17 The Friedmann equations ..... 181
17.1 Enter energy-momentum ..... 182
17.2 Enter thermodynamics ..... 183
17.3 Dust and radiation ..... 184
Exercises ..... 187
18 Universes of the past and future ..... 188
18.1 Spatially flat universes ..... 188
18.2 Curved universes with $\mathrm{\Lambda }=0$$\mathrm{\Lambda }=0$Lambda=0\Lambda=0$\mathrm{\Lambda }=0$ ..... 191
18.3 Einstein, Lemaître and Eddington ..... 192
18.4 A brief history of model universes ..... 196
Exercises ..... 199
19 Causality, infinity, and horizons ..... 201
19.1 Penrose diagrams ..... 202
19.2 The de Sitter spacetime ..... 209
19.3 Big-Bang singularities ..... 211
Exercises ..... 214
IV Orbits, stars, and black holes ..... 217
20 Newtonian orbits ..... 218
20.1 Kepler's laws ..... 219
20.2 Anatomy of an orbit ..... 220
20.3 Effective potentials ..... 222
20.4 Allowed trajectories ..... 223
20.5 The why? of orbits ..... 225
Exercises ..... 227
21 The Schwarzschild geometry ..... 229
21.1 Justifying the solution ..... 230
21.2 Components of the Riemann tensor ..... 231
21.3 A gravitating object ..... 232
21.4 The meaning of the coordinates ..... 234
Exercises ..... 235
22 Motion in the Schwarzschild geometry ..... 237
22.1 Constants of the motion ..... 238
22.2 Gravitational redshift ..... 239
22.3 Motion in Schwarzschild spacetime ..... 240
22.4 Example: the radial plunge ..... 242
Exercises ..... 244
23 Orbits in the Schwarzschild geometry ..... 246
23.1 Orbits for massive particles ..... 246
23.2 Stable circular orbits ..... 248
23.3 Precession of the perihelion ..... 249
Exercises ..... 252
24 Photons in the Schwarzschild geometry ..... 254
24.1 Photon trajectories ..... 254
24.2 Looking around ..... 258
Exercises ..... 261
25 Black holes ..... 262
25.1 The surface $r=2M$$r=2M$r=2Mr=2 M$r=2M$ ..... 264
25.2 The tortoise coordinate ..... 265
25.3 Death of an astronaut ..... 266
25.4 Looking around near a black hole ..... 267
25.5 Gravitational collapse ..... 268
Exercises ..... 270
26 Black-hole singularities ..... 272
26.1 Singularities ..... 272
26.2 Eddington-Finkelstein coordinates ..... 275
Exercises ..... 279
27 Kruskal-Szekeres coordinates ..... 280
27.1 Enter the Kruskal metric ..... 280
27.2 Wormholes ..... 284
27.3 Another Penrose diagram ..... 285
Exercises ..... 287
28 Hawking radiation ..... 289
28.1 Hawking radiation ..... 289
28.2 Black-hole thermodynamics ..... 292
Exercises ..... 296
29 Charged and rotating black holes ..... 297
29.1 Charged black holes ..... 297
29.2 Kerr black holes ..... 299
29.3 Interacting with the Kerr geometry ..... 304
Exercises ..... 306
V Geometry ..... 307
30 Classical curvature ..... 308
30.1 Curvature of a line ..... 308
30.2 Curvature with vectors ..... 310
30.3 Two-dimensional surfaces ..... 312
30.4 Gauss' equation ..... 314
30.5 Intrinsic and extrinsic curvature ..... 316
30.6 Riemann's project ..... 318
Exercises ..... 320
31 A reintroduction to geometry ..... 322
31.1 Old notions of vectors and gradients ..... 323
31.2 Vectors and vector fields ..... 324
31.3 Linear slot machines again ..... 327
31.4 Tensors again ..... 329
31.5 Examples of tensor operations ..... 330
Exercises ..... 332
32 Differential forms ..... 334
32.1 2-forms ..... 334
32.2 p-forms ..... 336
32.3 p-vectors ..... 337
Exercises ..... 339
33 Exterior and Lie derivatives ..... 340
33.1 Exterior calculus ..... 340
33.2 Commutators ..... 342
33.3 Lie derivatives of vectors ..... 344
33.4 Lie derivatives of tensors ..... 347
33.5 Killing vectors ..... 348
Exercises ..... 350
34 Geometry of the connection ..... 351
34.1 Covariant derivative in pictures ..... 352
34.2 Connection and exterior derivative ..... 353
34.3 Covariant derivative of tensors ..... 355
34.4 The metric revisited ..... 358
Exercises ..... 361
35 Riemann curvature revisited ..... 363
35.1 Geodesic deviation (slight return) ..... 363
35.2 Components of the curvature tensor ..... 366
35.3 Parallel transport again ..... 368
35.4 The meaning of the Ricci tensor ..... 370
Exercises ..... 372
36 Cartan's method ..... 374
36.1 Connection 1-forms ..... 374
36.2 Two rules ..... 377
36.3 Le repère mobile ..... 379
36.4 Example computations ..... 380
Exercises ..... 385
37 Duality and the volume form ..... 386
37.1 Motivation: 2-forms and flux ..... 386
37.2 Hodge star operation ..... 387
37.3 Volume forms ..... 392
Exercises ..... 395
38 Forms, chains, and Stokes' theorem ..... 397
38.1 Integration ..... 397
38.2 Integrating over forms ..... 400
38.3 Anatomy of an integral ..... 401
38.4 Boundaries and chains ..... 404
38.5 Stokes' theorem ..... 405
Exercises ..... 408
VI Classical and quantum fields ..... 411
39 Fluids as dry water ..... 412
39.1 Euler's equation ..... 413
39.2 Energy and Bernoulli's equation ..... 415
39.3 Energy-momentum tensor ..... 418
39.4 Relativistic fluids ..... 420
Exercises ..... 425
40 Lagrangian field theory ..... 428
40.1 Matter fields ..... 429
40.2 Action and equations of motion ..... 430
40.3 Fields in curved spacetime ..... 433
40.4 Motivating the Einstein equation ..... 434
40.5 Energy-momentum tensor ..... 437
40.6 Noether's theorem ..... 438
40.7 The perfect fluid ..... 440
Exercises ..... 443
41 Inflation ..... 445
41.1 Symmetry breaking ..... 446
41.2 Effective potentials ..... 449
41.3 Why flat? ..... 451
Exercises ..... 452
42 The electromagnetic field ..... 453
42.1 Electric charge in a field ..... 453
42.2 Faraday tensor and Maxwell equations ..... 455
42.3 Gauge freedom ..... 458
42.4 Geometrical electromagnetism ..... 460
Exercises ..... 464
43 Charge conservation and the Bianchi identity ..... 467
43.1 Conserving electric charge ..... 467
43.2 Electromagnetic gauge field ..... 469
43.3 Gravitational curvature ..... 471
Exercises ..... 475
44 Gauge fields ..... 476
44.1 Fibre bundles and gauge invariance ..... 476
44.2 Parallel transport and field strength ..... 480
Exercises ..... 483
45 Weak gravitational fields ..... 485
45.1 The Newtonian limit ..... 485
45.2 Linearized theory of gravitation ..... 487
45.3 Exploiting gauges ..... 488
Exercises ..... 492
46 Gravitational waves ..... 494
46.1 Waves in a gauge theory ..... 494
46.2 Lorenz gauge for gravitational waves ..... 496
46.3 Quadrupolar radiation ..... 501
46.4 Radiated energy and power ..... 503
46.5 An exact solution ..... 505
46.6 The discovery of gravitational waves ..... 506
Exercises ..... 509
47 The properties of gravitons ..... 512
47.1 Force-carrying particles ..... 512
47.2 Photon propagation and polarization ..... 514
47.3 Graviton propagation and polarization ..... 516
Exercises ..... 519
48 Higher dimensional spacetime ..... 520
48.1 Gauge transformations in five dimensions ..... 521
48.2 Unifying electromagnetism and gravitation ..... 522
Exercises ..... 525
49 From classical to quantum gravity ..... 527
49.1 Extra dimensions ..... 527
49.2 String theory ..... 530
49.3 Parametrizing the string ..... 532
49.4 Strings in relativity ..... 534
49.5 Superspace ..... 536
49.6 Loop quantum gravity ..... 537
49.7 Anti-de Sitter spacetime ..... 539
49.8 Our current best guess ..... 542
Exercises ..... 545
50 The Big-Bang singularity ..... 547
50.1 Facts about Euclidean geometry ..... 547
50.2 Orthogonal geodesics in spacetime ..... 548
50.3 Our Universe ..... 551
Exercises ..... 552
A Further reading ..... 554
B Conventions and notation ..... 562
B. 1 Electromagnetic units ..... 562
B. 2 Vectors, 1-forms and tensors ..... 562
B. 3 Covariant derivatives ..... 564
C Manifolds and bundles ..... 565
C. 1 Preliminaries ..... 566
C. 2 Maps and functions ..... 567
C. 3 One-to-one, into, and onto ..... 567
C. 4 Continuous maps ..... 568
C. 5 Manifolds, coordinates, and charts ..... 569
C. 6 Functions on the manifold ..... 571
C. 7 Differentiation on the manifold ..... 572
C. 8 Compact regions ..... 575
C. 9 Curves ..... 575
C. 10 Tangent spaces
578
578
C. 11 Fibre bundles ..... 578
D Embedding ..... 581
Exercises ..... 586
E Answers to selected problems ..... 587
Index ..... 614

Our Theory of Gravitation is as good as perfect: Lagrange, it is well known, has proved that the Planetary System, on this scheme, will endure forever; Laplace, still more cunningly, even guesses that it could not have been made on any other scheme.
Thomas Carlyle (1795-1881) Sartor Restartus
General relativity is one of the most profound statements in science. It is a theory of gravity that allows us to model the large-scale structure of the Universe; to understand and explain the workings of black holes; to reveal how gravity interacts with light waves and even how the Universe hosts its own, gravitational, waves. It is central to our notions of where the Universe comes from and what its eventual fate might be. The theory's conception was largely the work of one remarkable scientist. ${}^1$${}^1$^(1){ }^{1}${}^1$ General relativity is often viewed as a fearsomely difficult theory whose mastery is a rite of passage into the world of advanced physics. However, as we will show, the theory is based on simple principles which are straightforward to grasp. This initial chapter will outline the path we will take through the book and will introduce some important bits of jargon. We start with the word relativity.

Newton's ${}^2$${}^2$^(2){ }^{2}${}^2$ first law states that a body with no force acting on it will move in a straight line with a uniform velocity. This statement would be true if viewed in any inertial reference frame ('inertial' here means that the reference frame, which defines the coordinates used, is not accelerating). There are lots of inertial reference frames to choose from (all moving at different speeds and in different directions with respect to each other), but in all of them, Newton's first law holds. Even before Einstein came on the scene it was possible to formulate a principle of relativity:

Physical laws are the same in all inertial reference frames.
This implies that there is no absolute rest frame in Newtonian physics. ${}^3$${}^3$^(3){ }^{3}${}^3$ Any inertial reference frame will do, and we then have to describe motion relative to the inertial reference frame we have chosen.

${}^1$${}^1$^(1){ }^{1}${}^1$ Albert Einstein (1879-1955).
${}^3\mathrm{A}$${}^3\mathrm{A}$^(3)A{ }^{3} \mathrm{~A}${}^3\mathrm{A}$ rest frame of a particle is that frame of reference in which a particle is measured to be at rest

Fig. 1 Juggling is best performed in (a) an inertial reference frame, or one which is accelerating constantly, rather than (b) one which has a time-varying acceleration $a(t)$$a(t)$a(t)a(t)$a(t)$.

Example 0.1
Physical processes follow simple laws in inertial frames, because we can then apply Newton's laws in their simplest form.

The principle of relativity is, in effect, a symmetry principle. It tells us that physics works in the same way, however we choose our coordinates, as long as our coordinates are described relative to an inertial reference frame. We can transform from one inertial set of coordinates to another by rotating, translating, or even what we will call 'boosting'. A boost is a transformation to another coordinate system moving with uniform velocity with respect to the initial one.

An example of the independence of physics to boosts that is familiar to many is the sensation one experiences when seated on a train in a station and observing a neighbouring train moving forward. For a moment, you might think that your train is moving backward, and you need to check some fixed object on the station platform before you are sure which situation has occurred, and in effect whether your train is still in the station reference frame or in a new boosted reference frame.

The principle of relativity has been understood for a long time; Newton and Galileo accepted it. As we will explore in more detail in Chapter 1, Einstein's first revolutionary step, made in 1905, was to add an additional postulate:

The principle of invariant light speed:
As measured in any inertial reference frame, light propagates in empty space with a definite speed $c$$c$cc$c$, that is independent of the state of motion of the emitting body.

This principle has all manner of strange consequences that form the basis of Einstein's special theory of relativity. Why is it special? Because it is a theory that focuses on inertial reference frames and ignores gravity. Thus, it is restricted to some special (but important) cases. A good physical theory is said to be covariant if it transforms sensibly ${}^4$${}^4$^(4){ }^{4}${}^4$ under coordinate transformations. Special relativity is a theory which is covariant with respect to translations, rotations, reflections, and boosts. The boosts have to be carried out consistently with respect to the principle of invariant light speed and we will see in Chapter 1 that this must be carried out using a Lorentz transformation. Thus special relativity is said to be a theory which possesses Lorentz covariance.
Special relativity tells us that nothing can go faster than light. Thus, on a spacetime diagram, that is, a graph with time running up the page with spatial coordinates perpendicular, an observer's future and past can be represented as being inside a forward and backward light cone (see Fig. 2). Anything the observer can do now (throw a stone, shine a torch) can only influence the region of spacetime inside, or on, the forward light cone; anything that influences the observer now (the appearance of the night sky, an assassin's bullet) can only originate from inside, or on, the backward light cone. Moreover, if we populate spacetime with lots and lots of observers at different points, each will have their own light cone and all these light cones will be oriented in the same way [see Fig. 3(a)]. We shall see that this is a description of what is known as flat spacetime and is the situation that we assume to hold in special relativity.
(a)

(b)

Fig. 3 (a) The light cones in flat spacetime all line up at different points, like soldiers on parade. (b) The light cones in curved spacetime look much more disorderly, as if some of the soldiers on parade now have too much alcohol in their bloodstream.

This book is about Einstein's general theory of relativity in which gravity is described. To understand the significance of what Einstein did, it is helpful to first take a step back. Newton constructed a theory of gravity,
${}^4$${}^4$^(4){ }^{4}${}^4$ Clearly the notion of a 'sensible' transformation requires some explanation. For now, it can be thought of as the requirement that equations take the same form after transformation. This implies that no new terms should appear in an equation upon transformation to a different system of coordinates.

Fig. 2 The light cone in a spacetime diagram. Time is plotted vertically and the horizontal plane represents two of the three orthogonal spatial directions. An observer at the origin has the potential to influence any event inside her forward light cone and be influenced by any event inside her backward light cone.
${}^{5}G$${}^{5}G$^(5)G{ }^{5} G${}^{5}G$ is the gravitational constant $6.6741\times {10}^{-11}\mathrm{N}{\mathrm{kg}}^{-2}{\mathrm{m}}^2$$6.6741\times {10}^{-11}\mathrm{N}{\mathrm{kg}}^{-2}{\mathrm{m}}^2$6.6741 xx10^(-11)Nkg^(-2)m^(2)6.6741 \times 10^{-11} \mathrm{~N} \mathrm{~kg}^{-2} \mathrm{~m}^{2}$6.6741\times {10}^{-11}\mathrm{N}{\mathrm{kg}}^{-2}{\mathrm{m}}^2$, measured first by Henry Cavendish (1731-1810) in 1798.
${}^6$${}^6$^(6){ }^{6}${}^6$ Hence, the attitude of Thomas Carlyle in the quotation (written in 1836) that opened this chapter.
${}^7$${}^7$^(7){ }^{7}${}^7$ Newton knew this, as can be seen in the quotation heading the Preface to this book on page v. Newton has described the force produced by a distant mass, but a real force was felt to require a cause, and Newton couldn't come up with one. In the 1717 preface to his book on 'Opticks', he stated that he would not be taking gravity as an es would not be taking gravity as an es sential property of matter because he didn't know its cause because 1 am no yet satisfied about it for want of experiments'.
${}^8$${}^8$^(8){ }^{8}${}^8$ John Archibald Wheeler (1911-2008)
${}^9$${}^9$^(9){ }^{9}${}^9$ General relativity is a classical field theory. By classical we mean that the theory is not compatible with quantum mechanics. The search for a quantum theory of gravitation is still ongoing, a matter we will return to in Chapter 49
${}^{10}$${}^{10}$^(10){ }^{10}${}^{10}$ By matter fields we mean those fields describing massive particles or massive fluids, and also those describing phenomena such as electromagnetism, which is represented by a field with en ergetic, but massless, excitations.
published in his Principia in 1686, which meant that for the first time it was possible to appreciate that the same force that caused the Moon to orbit the Earth also caused the famous (and probably apocryphal) apple to fall from the tree. Newtonian gravity could be described by an equation, $F=GMm/r^2$$F=GMm/r^2$F=GMm//r^(2)F=G M m / r^{2}$F=GMm/r^2$, relating ${}^5$${}^5$^(5){ }^{5}${}^5$ the magnitude of the force $F$$F$FF$F$ between masses $M$$M$MM$M$ and $m$$m$mm$m$ separated by distance $r$$r$rr$r$. This inverse-square relationship beautifully explains the elliptical motions of the planets and led to many people thinking that gravity was a done deal. ${}^6$${}^6$^(6){ }^{6}${}^6$ However, there was a fly in the ointment. Newton had shown how gravity behaves, but he had not explained what it was. ${}^7$${}^7$^(7){ }^{7}${}^7$ Mechanical explanations were popular in the seventeenth century (it was, after all, the golden age of clockwork mechanisms) and in Newton's theory of gravity it was not possible to see where the gear wheels were located in this theory; there was no mechanism, no machinery, just an influence teleporting itself through empty space; it made no sense. What was transmitting the gravity through space? And what even was gravity anyway? It took Einstein's genius to realize that gravity isn't something that just gets transmitted through space. Space, or more accurately spacetime, is a structural property of the gravitational field, with the curvature in the very fabric of spacetime itself [see Fig. 3(b)] being directly determined by the matter within it. In the beautiful phrase coined by Wheeler ${}^8$${}^8$^(8){ }^{8}${}^8$ :

General relativity is a field theory that describes gravity. A field is a machine that takes a position in spacetime and outputs an object representing the amplitude of something at that point in spacetime. The amplitude could be a scalar, a vector, a tensor etc. ${}^9$${}^9$^(9){ }^{9}${}^9$ Field theories describe matter, such that we speak of the electromagnetic field as describing light and charges, of particle fields as describing elementary particles and of the fluid field as describing the dynamics of continuous fluids. General relativity tells us that the effects we call gravitational reflect the energy content of all of the matter fields ${}^{10}$${}^{10}$^(10){ }^{10}${}^{10}$ in the Universe. What makes general relativity unique as a field theory is that the energy of these matter fields, and hence gravitation itself, is inextricably linked to another, very special, field: the metric field that describes the geometry of space and time.
The clearest expression of how general relativity describes gravitation is the Einstein equation. This may be written conceptually as

The left-hand side of the Einstein equation is geometrical. The curvature is a geometrical property of space and time that follows from the metric field. The right-hand side of the Einstein equation is physical and reflects fields that describe the content of the Universe.

In formulating general relativity, Einstein began from this intuition, but initially struggled with the details of how curvature can be described
mathematically using geometrical techniques that were unfamiliar to him. In the century since Einstein's monumental work, there has been a great deal of progress in both the techniques and presentation of geometry, not least following the work of Élie Cartan, ${}^{11}$${}^{11}$^(11){ }^{11}${}^{11}$ but the reputation for difficulty that general relativity enjoys can still be traced back to the mathematical barrier this material presents to new students of gravitation. In fact, Einstein was helped by a friend, the mathematician Marcel Grossman, ${}^{12}$${}^{12}$^(12){ }^{12}${}^{12}$ to master geometry, but despite this, Einstein worked tirelessly for a further decade before the theory was complete. In this spirit of friendly help, the opening sections of this book are designed to help the gifted amateur understand the mathematical language of the lefthand (geometrical) side of the Einstein equation, but in due course we will fill in the details of both sides.

General relativity concerns the metric field, but what is that? The metric field can be thought of as a set of rules that allow us to work out the distances and angles between points in space and time. The geometrical description links space and time so inseparably that we refer to them as a single entity ${}^{13}$${}^{13}$^(13){ }^{13}${}^{13}$ spacetime. The metric field then describes geometry by providing the distances and angles between points in spacetime, known as events. The metric itself can be expressed by writing down the metric line element which is an equation for the interval between two closely spaced events. This allows us to carry out thought experiments where we imagine that spacetime has various particular curvatures and then investigate the consequences.
Ancient Alexandria's great mathematician Euclid ${}^{14}$${}^{14}$^(14){ }^{14}${}^{14}$ was never able to prove his parallel postulate: the intuition that two lines that start parallel will continue to be parallel out to infinity. It was realized by geometers in the eighteenth and nineteenth centuries that this is only true for a flat plane and that consistent geometries where parallel lines converge or diverge are possible in curved spaces. The deviation of parallel lines from parallelism gives us a test for, and measure of, curvature. In other words, any relative motion of two small, uncharged test particles, set off at the same speeds on parallel paths, must be the consequence of a gravitational field. The information about curvature is encoded in the metric. Einstein's equation is a differential equation that, when solved for a distribution of matter, gives us access to a metric field.
The metric is a field because, in general, it varies throughout spacetime. That is to say we insert a position in spacetime into the metric field and we are returned with a metric that allows us to compute the distance between events in that part of spacetime. The left-hand side of the Einstein equation can be thought of as a differential equation describing the variation of the metric field in spacetime and hence we obtain our notion of the curvature of spacetime [see Fig. 3(b)].
${}^{11}$${}^{11}$^(11){ }^{11}${}^{11}$ Élie Joseph Cartan (1869-1951).
${}^{12}$${}^{12}$^(12){ }^{12}${}^{12}$ Marcel Grossmann (1878-1936).
${}^{13}$${}^{13}$^(13){ }^{13}${}^{13}$ This notion of a single entity requires another conceptual leap: the coordinates in the metric are of no intrinsic significance. The symbol $t$$t$tt$t$, to which we have grown accustomed for representing time, becomes less important.
${}^{14}$${}^{14}$^(14){ }^{14}${}^{14}$ Euclid (who lived around 300 BC ).
${}^{15}$${}^{15}$^(15){ }^{15}${}^{15}$ Why? The Newtonian theory is known to provide a good description of gravitation in many of the circum stances in which we encounter it, i.e the limit of small gravitational interactions and of particles travelling slowly compared to the speed of light.

Fig. 4 The gravitational field $\overrightarrow{g}(\overrightarrow{r})$$\overrightarrow{g}(\overrightarrow{r})$vec(g)( vec(r))\vec{g}(\vec{r})$\overrightarrow{g}(\overrightarrow{r})$ around a particle of mass $M$$M$MM$M$.

Fig. 5 The gravitational potential $\mathrm{\Phi }(r)$$\mathrm{\Phi }(r)$Phi(r)\Phi(r)$\mathrm{\Phi }(r)$ (a scalar field) at a distance $r$$r$rr$r$ from a particle of mass $M$$M$MM$M$ at the origin.
${}^{16}$${}^{16}$^(16){ }^{16}${}^{16}$ Henry Cavendish used a torsion balance to measure the tiny gravitational attraction between lead spheres.

General relativity supersedes Newton's theory of gravity, but the two theories should agree if the gravitational fields are weak. ${}^{15}$${}^{15}$^(15){ }^{15}${}^{15}$ Therefore, it is worth restating the older Newtonian theory: Newton asserted that the force $\overrightarrow{F}$$\overrightarrow{F}$vec(F)\vec{F}$\overrightarrow{F}$ on a point mass $m$$m$mm$m$ at position $\overrightarrow{r}$$\overrightarrow{r}$vec(r)\vec{r}$\overrightarrow{r}$ due to a point mass $M$$M$MM$M$ at the origin is given by the vector equation

in which arrows denote three-dimensional vectors, and the minus sign expresses the fact that gravity is an attractive force. Since the force scales with the mass $m$$m$mm$m$, we can define a gravitational field vector $\overrightarrow{g}$$\overrightarrow{g}$vec(g)\vec{g}$\overrightarrow{g}$ as the force per unit mass, i.e.

and this is in fact a vector field $\overrightarrow{g}(\overrightarrow{r})$$\overrightarrow{g}(\overrightarrow{r})$vec(g)( vec(r))\vec{g}(\vec{r})$\overrightarrow{g}(\overrightarrow{r})$ that depends on position $\overrightarrow{r}$$\overrightarrow{r}$vec(r)\vec{r}$\overrightarrow{r}$. For a point mass, we then have (see Fig. 4)

The gravitational field is a conservative field of force (meaning the net work done in moving a point mass around a closed loop is zero, basically that the work done in rolling a ball up a hill is equivalent to the energy liberated when it rolls back down again), and hence we can write it as the gradient of a scalar potential. Conventionally, we include a minus sign and so write

where $\mathrm{\Phi }(\overrightarrow{r})$$\mathrm{\Phi }(\overrightarrow{r})$Phi( vec(r))\Phi(\vec{r})$\mathrm{\Phi }(\overrightarrow{r})$ is a scalar field known as the gravitational potential. For the case of a point mass $M$$M$MM$M$ at the origin, $\mathrm{\Phi }(\overrightarrow{r})=-GM/r$$\mathrm{\Phi }(\overrightarrow{r})=-GM/r$Phi( vec(r))=-GM//r\Phi(\vec{r})=-G M / r$\mathrm{\Phi }(\overrightarrow{r})=-GM/r$ (see Fig. 5). Gauss' theorem (to be discussed below) shows that if the mass at the origin is not point-like, but is spherically symmetric, then outside the radius of the mass distribution the same results still hold.

For a test mass on the surface of Earth, the gravitational force $F=mg$$F=mg$F=mgF=m g$F=mg$, where $g=$$g=$g=g=$g=$ $9.81\mathrm{m}{\mathrm{s}}^{-2}$$9.81\mathrm{m}{\mathrm{s}}^{-2}$9.81ms^(-2)9.81 \mathrm{~m} \mathrm{~s}^{-2}$9.81\mathrm{m}{\mathrm{s}}^{-2}$. Following the Cavendish experiment ${}^{16}$${}^{16}$^(16){ }^{16}${}^{16}$ of 1798 (and later improvement on it), the gravitational constant $G$$G$GG$G$ was measured to be $6.6741\times {10}^{-11}\mathrm{N}{\mathrm{kg}}^{-2}{\mathrm{m}}^2$$6.6741\times {10}^{-11}\mathrm{N}{\mathrm{kg}}^{-2}{\mathrm{m}}^2$6.6741 xx10^(-11)Nkg^(-2)m^(2)6.6741 \times 10^{-11} \mathrm{~N} \mathrm{~kg}^{-2} \mathrm{~m}^{2}$6.6741\times {10}^{-11}\mathrm{N}{\mathrm{kg}}^{-2}{\mathrm{m}}^2$. Cavendish described this experiment as 'weighing the world' because we can then use eqn 4 to deduce that

where $R_{\oplus }=6.378\times {10}^6\mathrm{m}$$R_{\oplus }=6.378\times {10}^6\mathrm{m}$R_(o+)=6.378 xx10^(6)mR_{\oplus}=6.378 \times 10^{6} \mathrm{~m}$R_{\oplus }=6.378\times {10}^6\mathrm{m}$ is the radius of the Earth. This then gives the mass of the Earth as $M_{\oplus }=5.97\times {10}^{24}\mathrm{kg}$$M_{\oplus }=5.97\times {10}^{24}\mathrm{kg}$M_(o+)=5.97 xx10^(24)kgM_{\oplus}=5.97 \times 10^{24} \mathrm{~kg}$M_{\oplus }=5.97\times {10}^{24}\mathrm{kg}$. Here we are using the commonly used symbol $\oplus$$\oplus$o+\oplus$\oplus$ to denote the Earth. With these two numbers, we can also work out the mean density of the Earth by dividing mass $M_{\oplus }$$M_{\oplus }$M_(o+)M_{\oplus}$M_{\oplus }$ by volume $\frac{4}{3}\pi R_{\oplus }^3$$\frac{4}{3}\pi R_{\oplus }^3$(4)/(3)piR_(o+)^(3)\frac{4}{3} \pi R_{\oplus}^{3}$\frac{4}{3}\pi R_{\oplus }^3$, which yields a value of $5.5\times {10}^3\mathrm{kg}{\mathrm{m}}^{-3}$$5.5\times {10}^3\mathrm{kg}{\mathrm{m}}^{-3}$5.5 xx10^(3)kgm^(-3)5.5 \times 10^{3} \mathrm{~kg} \mathrm{~m}^{-3}$5.5\times {10}^3\mathrm{kg}{\mathrm{m}}^{-3}$, just over a factor of 5 greater than water.

We can play a similar game with our nearest star, the Sun. The Earth's orbit around the Sun is elliptical, but it's not far from circular, so for an estimate we can equate the gravitational force on the Earth due to the Sun $G{M}_{\odot }{M}_{\oplus }/R_{\text{ES}}^2$$G{M}_{\odot }{M}_{\oplus }/R_{\text{ES }}^2$GM_(o.)M_(o+)//R_("ES ")^(2)G M_{\odot} M_{\oplus} / R_{\text {ES }}^{2}$G{M}_{\odot }{M}_{\oplus }/R_{\text{ES }}^2$ to the centripetal force $M_{\oplus }{v}^2/R_{\mathrm{ES}}$$M_{\oplus }{v}^2/R_{\mathrm{ES}}$M_(o+)v^(2)//R_(ES)M_{\oplus} v^{2} / R_{\mathrm{ES}}$M_{\oplus }{v}^2/R_{\mathrm{ES}}$, where $v$$v$vv$v$ is the speed of the Earth, $M_{\odot }$$M_{\odot }$M_(o.)M_{\odot}$M_{\odot }$ is the mass of the Sun ( $\odot$$\odot$o.\odot$\odot$ being the symbol we use for denoting the Sun) and $R_{\mathrm{ES}}$$R_{\mathrm{ES}}$R_(ES)R_{\mathrm{ES}}$R_{\mathrm{ES}}$, the separation of the Sun and Earth is called the astronomical unit (abbreviated A.U.). The value of $R_{\mathrm{ES}}$$R_{\mathrm{ES}}$R_(ES)R_{\mathrm{ES}}$R_{\mathrm{ES}}$ was first estimated by the Greeks by measuring the angle between a half-moon and the Sun (see Fig. 6), although subsequently improved in the seventeenth century and later by measuring the solar parallax using the transit of Venus. A modern value is $1.496\times {10}^{11}\mathrm{m}$$1.496\times {10}^{11}\mathrm{m}$1.496 xx10^(11)m1.496 \times 10^{11} \mathrm{~m}$1.496\times {10}^{11}\mathrm{m}$. The period $\tau$$\tau$tau\tau$\tau$ of the circular orbit is related to $v$$v$vv$v$ and $R_{\mathrm{ES}}$$R_{\mathrm{ES}}$R_(ES)R_{\mathrm{ES}}$R_{\mathrm{ES}}$ by $\tau =2\pi R_{\mathrm{ES}}/v$$\tau =2\pi R_{\mathrm{ES}}/v$tau=2piR_(ES)//v\tau=2 \pi R_{\mathrm{ES}} / v$\tau =2\pi R_{\mathrm{ES}}/v$, where our equating of centripetal and gravitational forces yields $v^2=G{M}_{\odot }/R_{\mathrm{ES}}$$v^2=G{M}_{\odot }/R_{\mathrm{ES}}$v^(2)=GM_(o.)//R_(ES)v^{2}=G M_{\odot} / R_{\mathrm{ES}}$v^2=G{M}_{\odot }/R_{\mathrm{ES}}$. We can hence deduce from $\tau =1$$\tau =1$tau=1\tau=1$\tau =1$ year that $M_{\odot }=1.99\times {10}^{30}\mathrm{kg}$$M_{\odot }=1.99\times {10}^{30}\mathrm{kg}$M_(o.)=1.99 xx10^(30)kgM_{\odot}=1.99 \times 10^{30} \mathrm{~kg}$M_{\odot }=1.99\times {10}^{30}\mathrm{kg}$. The density of the Sun, using $R_{\odot }=6.96\times {10}^8\mathrm{m}$$R_{\odot }=6.96\times {10}^8\mathrm{m}$R_(o.)=6.96 xx10^(8)mR_{\odot}=6.96 \times 10^{8} \mathrm{~m}$R_{\odot }=6.96\times {10}^8\mathrm{m}$, then works out to be around $1.4\times {10}^3\mathrm{kg}{\mathrm{m}}^{-3}$$1.4\times {10}^3\mathrm{kg}{\mathrm{m}}^{-3}$1.4 xx10^(3)kgm^(-3)1.4 \times 10^{3} \mathrm{~kg} \mathrm{~m}^{-3}$1.4\times {10}^3\mathrm{kg}{\mathrm{m}}^{-3}$, just a bit larger than that of water.
One conclusion from all of this is that, from a nineteenth-century perspective, the idea of a black hole (an object with such intense surface gravity that even light could not escape) seems highly unlikely. The escape velocity $v_{\text{esc}}$$v_{\text{esc }}$v_("esc ")v_{\text {esc }}$v_{\text{esc }}$ from a spherical object of radius $R$$R$RR$R$, mass $M=\frac{4}{3}\pi \rho R^3$$M=\frac{4}{3}\pi \rho R^3$M=(4)/(3)pi rhoR^(3)M=\frac{4}{3} \pi \rho R^{3}$M=\frac{4}{3}\pi \rho R^3$ and density $\rho$$\rho$rho\rho$\rho$ is simply worked out by equating the kinetic energy $\frac{1}{2}m{v}_{\text{esc}}^2$$\frac{1}{2}m{v}_{\text{esc }}^2$(1)/(2)mv_("esc ")^(2)\frac{1}{2} m v_{\text {esc }}^{2}$\frac{1}{2}m{v}_{\text{esc }}^2$ of a launching test mass to the depth $m|\mathrm{\Phi }|=GMm/R$$m|\mathrm{\Phi }|=GMm/R$m|Phi|=GMm//Rm|\Phi|=G M m / R$m|\mathrm{\Phi }|=GMm/R$, of the potential energy well in which it starts its journey. This yields $v_{\text{esc}}=\sqrt{2GM/R}=$$v_{\text{esc }}=\sqrt{2GM/R}=$v_("esc ")=sqrt(2GM//R)=v_{\text {esc }}=\sqrt{2 G M / R}=$v_{\text{esc }}=\sqrt{2GM/R}=$ $R\sqrt{8\pi G\rho }$$R\sqrt{8\pi G\rho }$Rsqrt(8pi G rho)R \sqrt{8 \pi G \rho}$R\sqrt{8\pi G\rho }$. This result scales linearly with $R$$R$RR$R$ and would reach $v_{\mathrm{esc}}=c$$v_{\mathrm{esc}}=c$v_(esc)=cv_{\mathrm{esc}}=c$v_{\mathrm{esc}}=c$ only when

Since the best-studied objects in the Universe were those in our own Solar System, and these have mean densities that don't exceed that of water by more than a factor of about five, and since the rest of the Universe seems to be filled with stars that look somewhat similar to the Sun, then eqn 7 would only be likely to be satisfied by an object with radius of more than an astronomical unit, the distance from the Earth to the Sun. No normal stars were thought to be this big. Thus, it didn't seem as if eqn 7 would hold. ${}^{17}$${}^{17}$^(17){ }^{17}${}^{17}$

Because $\overrightarrow{g}$$\overrightarrow{g}$vec(g)\vec{g}$\overrightarrow{g}$ points inwards to any point mass $M$$M$MM$M$ at the origin, we deduce that the integral of $\overrightarrow{g}$$\overrightarrow{g}$vec(g)\vec{g}$\overrightarrow{g}$ over any spherical surface $S$$S$SS$S$ of radius $R$$R$RR$R$ surrounding the origin is

The divergence theorem is a result from vector calculus and says that the left-hand side of this equation, a surface integral of the flux of the vector $\overrightarrow{g}$$\overrightarrow{g}$vec(g)\vec{g}$\overrightarrow{g}$ out of the surface, can be rewritten as an integral over the volume of the divergence of $\overrightarrow{g}$$\overrightarrow{g}$vec(g)\vec{g}$\overrightarrow{g}$, written as $\overrightarrow{\mathrm{\nabla }}\cdot \overrightarrow{g}$$\overrightarrow{\mathrm{\nabla }}\cdot \overrightarrow{g}$vec(grad)* vec(g)\vec{\nabla} \cdot \vec{g}$\overrightarrow{\mathrm{\nabla }}\cdot \overrightarrow{g}$. Hence, we have

where here the volume element $\mathrm{d}V={\mathrm{d}}^{3}r$$\mathrm{d}V={\mathrm{d}}^{3}r$dV=d^(3)r\mathrm{d} V=\mathrm{d}^{3} r$\mathrm{d}V={\mathrm{d}}^{3}r$, i.e. the gravitational flux out of a surface is equal to the integral of the divergence of the gravitational field inside the volume enclosed by the surface. From this, we can deduce that ${}^{18}$${}^{18}$^(18){ }^{18}${}^{18}$

The Newtonian results for a point mass can be generalized for the field due to a distribution of mass since Newtonian theory is linear. Thus, for example,

Fig. 6 Diagram (not to scale) showing how the distance to the Sun can be estimated by measuring the angle between the Sun and the half-Moon. The distances are given in A.U. The calculation relies on an estimate of the distance to the Moon which can be estimated from measurements of lunar parallax.

Mass: $M_{\oplus }=5.97\times {10}^{24}\mathrm{kg}$$M_{\oplus }=5.97\times {10}^{24}\mathrm{kg}$M_(o+)=5.97 xx10^(24)kgM_{\oplus}=5.97 \times 10^{24} \mathrm{~kg}$M_{\oplus }=5.97\times {10}^{24}\mathrm{kg}$ Radius: $R_{\oplus }=6.378\times {10}^6\mathrm{m}$$R_{\oplus }=6.378\times {10}^6\mathrm{m}$R_(o+)=6.378 xx10^(6)mR_{\oplus}=6.378 \times 10^{6} \mathrm{~m}$R_{\oplus }=6.378\times {10}^6\mathrm{m}$

The Sun ${}^{-}$${}^{-}$^(-){ }^{-}${}^{-}$
Mass: $M_{\odot }=1.99\times {10}^{30}\mathrm{kg}$$M_{\odot }=1.99\times {10}^{30}\mathrm{kg}$M_(o.)=1.99 xx10^(30)kgM_{\odot}=1.99 \times 10^{30} \mathrm{~kg}$M_{\odot }=1.99\times {10}^{30}\mathrm{kg}$ Radius: $R_{\odot }=6.96\times {10}^8\mathrm{m}$$R_{\odot }=6.96\times {10}^8\mathrm{m}$R_(o.)=6.96 xx10^(8)mR_{\odot}=6.96 \times 10^{8} \mathrm{~m}$R_{\odot }=6.96\times {10}^8\mathrm{m}$

${}^{17}$${}^{17}$^(17){ }^{17}${}^{17}$ As we shall see later, it is possible to have compact objects such as neutron stars which have enormous densities. This only became possible to understand after the development of quantum mechanics.
${}^{18}$${}^{18}$^(18){ }^{18}${}^{18}$ The Dirac delta function $\delta (x)$$\delta (x)$delta(x)\delta(x)$\delta (x)$ is a function localized at the origin and which has integral unity. It is the perfect model of a localized particle, and is used here to fix the point mass $M$$M$MM$M$ at the origin. We have written a three-dimensional delta function $\delta (\overrightarrow{r})\equiv \delta (x)\delta (y)\delta (z)$$\delta (\overrightarrow{r})\equiv \delta (x)\delta (y)\delta (z)$delta( vec(r))-=delta(x)delta(y)delta(z)\delta(\vec{r}) \equiv \delta(x) \delta(y) \delta(z)$\delta (\overrightarrow{r})\equiv \delta (x)\delta (y)\delta (z)$, often denoted ${\delta }^{(3)}(\overrightarrow{x})$${\delta }^{(3)}(\overrightarrow{x})$delta^((3))( vec(x))\delta^{(3)}(\vec{x})${\delta }^{(3)}(\overrightarrow{x})$. The integral of a $d$$d$dd$d$-dimensional Dirac delta function ${\delta }^{(d)}(\overrightarrow{x})$${\delta }^{(d)}(\overrightarrow{x})$delta^((d))( vec(x))\delta^{(d)}(\vec{x})${\delta }^{(d)}(\overrightarrow{x})$ is given by

It is defined by

${}^{19}$${}^{19}$^(19){ }^{19}${}^{19}$ An arbitrary distribution of mass can be written as an integral of point masses using

The integral form of eqn 12 is

where $M=\int \rho \left({\overrightarrow{r}}^{\prime }\right){\mathrm{d}}^3{r}^{\prime }$$M=\int \rho \left({\overrightarrow{r}}^{\prime }\right){\mathrm{d}}^3{r}^{\prime }$M=int rho( vec(r)^('))d^(3)r^(')M=\int \rho\left(\vec{r}^{\prime}\right) \mathrm{d}^{3} r^{\prime}$M=\int \rho \left({\overrightarrow{r}}^{\prime }\right){\mathrm{d}}^3{r}^{\prime }$ is the total mass enclosed inside the surface $S$$S$SS$S$, This result, which generalizes eqn 8 to an arbitrary distribution of mass, is often known as Gauss' theorem for gravitational fields.
${}^{20}$${}^{20}$^(20){ }^{20}${}^{20}$ In electrostatics, the force on charge $q$$q$qq$q$ is $\overrightarrow{F}=q\overrightarrow{E}$$\overrightarrow{F}=q\overrightarrow{E}$vec(F)=q vec(E)\vec{F}=q \vec{E}$\overrightarrow{F}=q\overrightarrow{E}$ where $\overrightarrow{E}=-\overrightarrow{\mathrm{\nabla }}\varphi$$\overrightarrow{E}=-\overrightarrow{\mathrm{\nabla }}\varphi$vec(E)=- vec(grad)phi\vec{E}=-\vec{\nabla} \phi$\overrightarrow{E}=-\overrightarrow{\mathrm{\nabla }}\varphi$ is the electric field and $\varphi$$\varphi$phi\phi$\varphi$ is the electrostatic potential. Gauss' theorem for electrostatics is (in SI units)

where $Q$$Q$QQ$Q$ is the charge enclosed by the surface $S$$S$SS$S$, and $\overrightarrow{\mathrm{\nabla }}\cdot \overrightarrow{E}=\rho /{ϵ}_0$$\overrightarrow{\mathrm{\nabla }}\cdot \overrightarrow{E}=\rho /{ϵ}_0$vec(grad)* vec(E)=rho//epsilon_(0)\vec{\nabla} \cdot \vec{E}=\rho / \epsilon_{0}$\overrightarrow{\mathrm{\nabla }}\cdot \overrightarrow{E}=\rho /{ϵ}_0$ where $\rho$$\rho$rho\rho$\rho$ here is the charge density, and

is Poisson's equation.
${}^{21}$${}^{21}$^(21){ }^{21}${}^{21}$ Carlyle may have said that 'Our Theory of Gravitation is as good as perfect...' in the quote that opened the chapter, but this discrepancy turned out to be rather significant!
is the gravitational potential at position $\overrightarrow{r}$$\overrightarrow{r}$vec(r)\vec{r}$\overrightarrow{r}$ from a distribution of masses with density $\rho \left({\overrightarrow{r}}^{\prime }\right)$$\rho \left({\overrightarrow{r}}^{\prime }\right)$rho( vec(r)^('))\rho\left(\vec{r}^{\prime}\right)$\rho \left({\overrightarrow{r}}^{\prime }\right)$. The divergence of the gravitational field can then be written (generalizing eqn 12) as ${}^{19}$${}^{19}$^(19){ }^{19}${}^{19}$

Equivalently, this can be written using the gravitational potential $\mathrm{\Phi }$$\mathrm{\Phi }$Phi\Phi$\mathrm{\Phi }$, via $\overrightarrow{g}=-\overrightarrow{\mathrm{\nabla }}\mathrm{\Phi }$$\overrightarrow{g}=-\overrightarrow{\mathrm{\nabla }}\mathrm{\Phi }$vec(g)=- vec(grad)Phi\vec{g}=-\vec{\nabla} \Phi$\overrightarrow{g}=-\overrightarrow{\mathrm{\nabla }}\mathrm{\Phi }$, to yield

which is analogous to Poisson's equation in electrostatics. ${}^{20}$${}^{20}$^(20){ }^{20}${}^{20}$
The dimensions of the gravitational potential $\mathrm{\Phi }$$\mathrm{\Phi }$Phi\Phi$\mathrm{\Phi }$ are (velocity) ${}^2$${}^2$^(2){ }^{2}${}^2$ so one might wonder what happens when $|\mathrm{\Phi }|$$|\mathrm{\Phi }|$|Phi||\Phi|$|\mathrm{\Phi }|$ becomes of the same order as $c^2$$c^2$c^(2)c^{2}$c^2$, where $c$$c$cc$c$ is the speed of light. This would be equivalent to the size of the gravitational potential energy $m|\mathrm{\Phi }|$$m|\mathrm{\Phi }|$m|Phi|m|\Phi|$m|\mathrm{\Phi }|$ of a mass $m$$m$mm$m$ becoming of the same order as the rest mass energy $m{c}^2$$m{c}^2$mc^(2)m c^{2}$m{c}^2$. This gives a rough criterion for when Newton's law of gravitation is likely to break down and the effects of general relativity to become extremely important. However, as we shall see in this book, the effects of general relativity can become significantly important, even before this point is reached.

As with our earlier book on quantum field theory, our imagined reader is an amateur. We have written this book for someone wanting to learn general relativity without (at least initially) joining the ranks of professional relativists; but (s)he is gifted, possessing a curious and adaptable mind and willing to embark on a significant intellectual challenge; (s)he has abundant curiosity about the physical world, a basic grounding in undergraduate physics, and a desire to be told an entertaining and intellectually stimulating story, but will not feel patronized if a few
mathematical niceties are spelled out in detail. We appreciate that some readers will want to get to the physical predictions of the theory as soon as possible, as their primary concern is with understanding what the Universe is actually like. Others will have more interest in the mathematical structure of the theory; such readers will want to know more about how some more advanced geometric formalism can yield additional insights. We have tried to cater for both types of readers and have designed the book so that it is possible to dip in and out of sections that may be more or less to a reader's taste, though we recommend all beginners to persevere with at least the first thirteen chapters.
The book is structured as follows. We begin in Part I with an introduction to the geometry of flat spacetime, reviewing special relativity and setting up the mathematics of the metric. Part II introduces the mathematics of curvature and sets up the physics of general relativity and finishes with the Einstein field equation. Part III applies these ideas to the Universe and studies various models used in cosmology. Part IV turns to smaller structures inside the Universe: stars, black holes and their orbits. Part V contains a more formal treatment of geometry which may be of more interest to those with more mathematical inclinations. Part VI considers general relativity as a type of field theory and examines how one might link the ideas in our best theory of gravitation to our most successful theories of quantum fields. Before we get going, we will say a few words about units.

Most readers will be familiar with SI units and we will begin the book using them. However, once we get going, we will switch over to what are known as geometrized units in which we set $G=c=1.{}^{22}$$G=c=1.{}^{22}$G=c=1.^(22)G=c=1 .{ }^{22}$G=c=1.{}^{22}$ This has the great advantage of simplifying equations into more memorable forms since they will no longer be encumbered with unnecessary factors of $c$$c$cc$c$ and $G$$G$GG$G$ whose presence, to the experts, is 'obvious'. It of course has the great disadvantage of creating some confusion whenever a numerical result it needed, but after a bit of practice this does become second nature. Because of the potential confusion for newcomers to the field, we will frequently translate back to SI units (which we will tend to call 'real-world' units) when we need to. Here is an explanation of how to translate between the two systems.

Conversion factors to convert from quantities expressed in real-world units into geometrized units can be computed by noting that the dimension ${}^{23}$${}^{23}$^(23){ }^{23}${}^{23}$ of $c$$c$cc$c$ is $\mathrm{L}/\mathrm{T}$$\mathrm{L}/\mathrm{T}$L//T\mathrm{L} / \mathrm{T}$\mathrm{L}/\mathrm{T}$ in the real world, while the dimension of $G/c^2$$G/c^2$G//c^(2)G / c^{2}$G/c^2$ is $\mathrm{L}/\mathrm{M}$$\mathrm{L}/\mathrm{M}$L//M\mathrm{L} / \mathrm{M}$\mathrm{L}/\mathrm{M}$. To convert a quantity with realworld dimension time into geometrized units, multiply by $c$$c$cc$c$. To convert a quantity with real-world dimension mass multiply by $G/c^2$$G/c^2$G//c^(2)G / c^{2}$G/c^2$. Both of these quantities then have units of length in the geometrized system.
$\curvearrowright$$\curvearrowright$↷\curvearrowright$\curvearrowright$ It is certainly not necessary to read the book in order In fact, we would recommend skipping several sections on a first reading. Boxes like this one are intended to allow you to navigate a path through the text.
${}^{22}$${}^{22}$^(22){ }^{22}${}^{22}$ The reader will be let in gently to geometrized units. We will not begin to ometrized units. We will not begin to
set $c=1$$c=1$c=1c=1$c=1$ until Chapter 2, and will not set $G=1$$G=1$G=1G=1$G=1$ as well until Part III. In addition, Appendix B contains a summary of the units we use to discuss electromagnetism, along with a summary of useful notation.
${}^{23}$${}^{23}$^(23){ }^{23}${}^{23}$ We denote dimension of length by L , time by T and mass by M .

The generalized version of the above argument says that if a quantity has units ${\mathrm{L}}^n{\mathrm{T}}^m{\mathrm{M}}^p$${\mathrm{L}}^n{\mathrm{T}}^m{\mathrm{M}}^p$L^(n)T^(m)M^(p)\mathrm{L}^{n} \mathrm{~T}^{m} \mathrm{M}^{p}${\mathrm{L}}^n{\mathrm{T}}^m{\mathrm{M}}^p$ in the real world, then it has units ${\mathrm{L}}^{n+m+p}$${\mathrm{L}}^{n+m+p}$L^(n+m+p)\mathrm{L}^{n+m+p}${\mathrm{L}}^{n+m+p}$ in the geometrized system and the conversion factor is $c^m{\left(G/c^2\right)}^p$$c^m{\left(G/c^2\right)}^p$c^(m)(G//c^(2))^(p)c^{m}\left(G / c^{2}\right)^{p}$c^m{\left(G/c^2\right)}^p$. The table gives some examples.

If you want to convert an equation expressed in geometrized units into real-world units, multiply the quantities in the table by their respective conversion factors.

In geometrized units, the Einstein equation is $\mathit{G}=8\pi \mathit{T}$$\mathit{G}=8\pi \mathit{T}$G=8pi T\boldsymbol{G}=8 \pi \boldsymbol{T}$\mathit{G}=8\pi \mathit{T}$, where $\mathit{G}$$\mathit{G}$G\boldsymbol{G}$\mathit{G}$ and $\mathit{T}$$\mathit{T}$T\boldsymbol{T}$\mathit{T}$ are tensors that we will define later in the book (and should not be confused with the gravitational constant $G$$G$GG$G$ and temperature $T$$T$TT$T$ ). The left-hand side of the Einstein equation has units ${\mathrm{L}}^{-2}$${\mathrm{L}}^{-2}$L^(-2)\mathrm{L}^{-2}${\mathrm{L}}^{-2}$, and the right-hand side has units of energy density ( ${\mathrm{L}}^{-1}{\mathrm{T}}^{-2}\mathrm{M}$${\mathrm{L}}^{-1}{\mathrm{T}}^{-2}\mathrm{M}$L^(-1)T^(-2)M\mathrm{L}^{-1} \mathrm{~T}^{-2} \mathrm{M}${\mathrm{L}}^{-1}{\mathrm{T}}^{-2}\mathrm{M}$ ). The left-hand side is multiplied by unity; the right by $G/c^4$$G/c^4$G//c^(4)G / c^{4}$G/c^4$ and we obtain